Self-intersection numbers of curves in the doubly-punctured plane

TL;DR

This paper establishes exact bounds for the minimal self-intersection number of curves in the doubly-punctured plane based on their combinatorial length, providing sharp bounds and identifying classes that attain these bounds.

Contribution

It derives precise upper and lower bounds for self-intersection numbers in the doubly-punctured plane, including sharp bounds and classes attaining these bounds, advancing understanding of curve complexity.

Findings

Upper bound for self-intersection number: L^2/4 + L/2 - 1

Sharp bounds for even and odd lengths, with classes attaining these bounds

Lower bounds for self-intersection number: L/2 - 1 (even), (L - 1)/2 (odd)

Abstract

We address the problem of computing bounds for the self-intersection number (the minimum number of self-intersection points) of members of a free homotopy class of curves in the doubly-punctured plane as a function of their combinatorial length L; this is the number of letters required for a minimal description of the class in terms of the standard generators of the fundamental group and their inverses. We prove that the self-intersection number is bounded above by L^2/4 + L/2 - 1, and that when L is even, this bound is sharp; in that case there are exactly four distinct classes attaining that bound. When L is odd, we establish a smaller, conjectured upper bound ((L^2 - 1)/4)) in certain cases; and there we show it is sharp. Furthermore, for the doubly-punctured plane, these self-intersection numbers are bounded below, by L/2 - 1 if L is even, (L - 1)/2 if L is odd; these bounds are sharp.